Hierarchical and Higher-Order Factor Structures in the Rasch Tradition: A Didactic.

In this paper, we consider hierarchical and higher-order factor models and the relationship between them, and, in particular, we use Rasch models to focus on the exploration of these models. We present these models, their similarities and/or differences from within the Rasch modeling perspective and discuss their use in various settings. One motivation for this work is that certain well-known similarities and differences between the equivalent models in the two-parameter logistic model (2PL) approach do not apply in the Rasch modeling tradition. Another motivation is that there is some ambiguity as to the potential uses of these models, and we seek to clarify those uses. In recent work in the Item Response Theory (IRT) literature, the estimation of these models has been mostly presented using the Bayesian framework: here we show the use of these models using traditional maximum likelihood methods. We also show how to re-parameterize these models, which in some cases can improve estimation and convergence. These alternative parameterizations are also useful in "translating" suggestions for the 2PL models to the Rasch tradition (since these suggestions involve the interpretation of item discriminations, which are required to be unity in the Rasch tradition). Alternative parameterizations can also be used to clarify the relationship among these models. We discuss the use of these models for modeling multidimensionality and testlet effects and compare the interpretation of the obtained solutions to the interpretation for the multidimenisional Rasch model - a more common approach for accounting multidimensionality in the Rasch tradition. We demonstrate the use of these models using the partial credit model.